A bathroom door that swings into the room and locks from the inside can turn a slow slip into a long wait on the floor. The sizing question is simple: can Dad get in and out with a walker, and can someone get in from the hall without a fight over the knob? Work from the door you have, not the one you wish you had.
Local option: New Plan Care gives families in the St. Louis area in-home help so a parent can stay home: companionship, help with bathing and dressing, meals, and overnight care. Call (314) 405-0887.
Measure the opening, not the door
Measure the clear width between the door stop and the edge of the open door, not the width of the door slab itself. Then measure the widest point of his walker or cane setup. The walker needs to pass with room on both sides, and a grab bar or a wall-mounted frame can eat that margin without anyone noticing.
Check the swing and the space it needs
An inward-swinging door takes the floor space behind it. Stand where Dad would stand after a fall, close the door, and see whether it can open fully without hitting the toilet, the tub, or a fixed fixture. If it cannot, the swing is the real problem and the latch is secondary.
Set the latch to a release that works from outside
Privacy latches come in push-button, slide, and turn-knob styles. Check each one for a release slot or a small hole on the outside of the door that a thin tool will reach. A latch with no outside release is a lock, and the room needs a way in that does not depend on the person inside. Test it with the door shut, from the hall, using the tool you would really use.
What to compare before you change anything
Compare three things: the swing direction, the clear width against the walker, and the outside release. A door that swings out may need its hinges changed, and a rented unit may need the landlord's sign-off first, so check that before you buy parts. Do not accept any latch that cannot be opened from the hall. Dad will likely push back on a new latch, so present it as a release rather than a lock.
Your next move: tonight, close the bathroom door, latch it the way Dad does, and time how long it takes to open from the hall with the tool you would use. Then take the walker through the opening with the door fully open and note the gap on each side. Write both numbers on the door frame in pencil. If the release is not quick to reach or the walker scrapes the frame, call a handyman about the hinge side or the latch, not about a new lock.